Dihedral angles of 1-α,-β-R-5-O-methanesulfonyl-2,3-O-isopropilidene-L-lyxose calculated from NMR data with 3-sphere approach and Lambert-Wu R value

Abstract

3-Sphere and Lie group theories (Fig. 1, 2) enable calculations of the dihedral angle with right sign and stereochemistry from NMR data[1, 2]: carbon, proton chemical shift ( 1 H, 13 C [ppm]), and vicinal constant coupling ( 3 J HH [Hz]), and manifold equations transform NMR data in angles. The dihedral angles of the mixture 1-α:1-β-L-lyxose are calculated with Lambert R value under 3-sphere approach rule from the ratio of α H1 -H 3 H 4 :β H1 -H 3 H 4 , or with manifold equation from carbon or proton chemical shift under conic section. Six sets angles on two units, U1 and its parallel unit S1, are build with manifold angles under equilateral triangle rule. The coordinates[3] on R 4 (X 0 = cosϕsinθ HnHn+1 , X 1 = sinϕsinξ*/2, X 2 = cosθ HnHn+1 sinϕ, X 3 = sinθ HnHn+1 sinξ/2), confirmed by algebraic equations (cis-, trans-ee: 3 J HH = ϕ 1/2 /2, trans-aa: 3 J HH = ϕ 1/2 ), gives greet relationships between the dihedral angle and the vicinal angle. Scoop of the paper: Our aim is to demonstrated once again the gift of NMR data for calculation dihedral angles with 3-sphere approach. In the absence of one of the NMR data the manifold equations are solutions for all cases.

Hopf Fibration on Relationship Between Dihedral Angle θ<sub>HnHn+1</sub>[deg] and Vicinal Angle ϕ[deg], Angles Calculated from NMR Data with 3-Shere Approach and Java Script

“…5, enable the calculation of the dihedral angle from chemical shift (carbon and/or proton) and vicinal coupling constant. The advantages of the 3-sphere approach [19]: the right stereochemistry with corresponding sign results from a perfect mathematic symmetry, gives by nature itself (the universe shape), with applications on conformation [16,22] and configuration [23] of iminocyclitols [24]. A method for calculation of the dihedral angles from NMR data by hand with right stereochemistry and sign, or with Java Script programs [25,26]: 1.…”

“…In step 2 [19] are calculated angles of set A and B from the differences between two atoms of carbon chemical shift δ CnCn+1 [ppm], or two protons δ CnCn+1 [ppm], with the manifold equations (sin -1 R m = θ An , n = 1-6): conic section (R m = f(∆δ XnXn+1 , X = C, H)), and skew circles -non coplanar Villarceau circles (R m = f(∆δ HnHn+1 / ∆δ CnCn+1 )).…”